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Theorem hofpropd 17115
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
hofpropd.1 (𝜑 → (Homf𝐶) = (Homf𝐷))
hofpropd.2 (𝜑 → (compf𝐶) = (compf𝐷))
hofpropd.c (𝜑𝐶 ∈ Cat)
hofpropd.d (𝜑𝐷 ∈ Cat)
Assertion
Ref Expression
hofpropd (𝜑 → (HomF𝐶) = (HomF𝐷))

Proof of Theorem hofpropd
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofpropd.1 . . 3 (𝜑 → (Homf𝐶) = (Homf𝐷))
21homfeqbas 16563 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
32sqxpeqd 5281 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
43adantr 466 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
5 eqid 2771 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
6 eqid 2771 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
7 eqid 2771 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
81adantr 466 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (Homf𝐶) = (Homf𝐷))
9 xp1st 7347 . . . . . . 7 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑦) ∈ (Base‘𝐶))
109ad2antll 708 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (1st𝑦) ∈ (Base‘𝐶))
11 xp1st 7347 . . . . . . 7 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑥) ∈ (Base‘𝐶))
1211ad2antrl 707 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (1st𝑥) ∈ (Base‘𝐶))
135, 6, 7, 8, 10, 12homfeqval 16564 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) = ((1st𝑦)(Hom ‘𝐷)(1st𝑥)))
14 xp2nd 7348 . . . . . . . 8 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑥) ∈ (Base‘𝐶))
1514ad2antrl 707 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (2nd𝑥) ∈ (Base‘𝐶))
16 xp2nd 7348 . . . . . . . 8 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑦) ∈ (Base‘𝐶))
1716ad2antll 708 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (2nd𝑦) ∈ (Base‘𝐶))
185, 6, 7, 8, 15, 17homfeqval 16564 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) = ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))
1918adantr 466 . . . . 5 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ 𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥))) → ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) = ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))
205, 6, 7, 8, 12, 15homfeqval 16564 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐷)(2nd𝑥)))
21 df-ov 6796 . . . . . . . . 9 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
22 df-ov 6796 . . . . . . . . 9 ((1st𝑥)(Hom ‘𝐷)(2nd𝑥)) = ((Hom ‘𝐷)‘⟨(1st𝑥), (2nd𝑥)⟩)
2320, 21, 223eqtr3g 2828 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩) = ((Hom ‘𝐷)‘⟨(1st𝑥), (2nd𝑥)⟩))
24 1st2nd2 7354 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2524ad2antrl 707 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2625fveq2d 6336 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
2725fveq2d 6336 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐷)‘𝑥) = ((Hom ‘𝐷)‘⟨(1st𝑥), (2nd𝑥)⟩))
2823, 26, 273eqtr4d 2815 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐷)‘𝑥))
2928adantr 466 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐷)‘𝑥))
30 eqid 2771 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
31 eqid 2771 . . . . . . . 8 (comp‘𝐷) = (comp‘𝐷)
328ad2antrr 705 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (Homf𝐶) = (Homf𝐷))
33 hofpropd.2 . . . . . . . . 9 (𝜑 → (compf𝐶) = (compf𝐷))
3433ad3antrrr 709 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (compf𝐶) = (compf𝐷))
3510ad2antrr 705 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st𝑦) ∈ (Base‘𝐶))
3612ad2antrr 705 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st𝑥) ∈ (Base‘𝐶))
3717ad2antrr 705 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd𝑦) ∈ (Base‘𝐶))
38 simplrl 762 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)))
3925ad2antrr 705 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
4039oveq1d 6808 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑥(comp‘𝐶)(2nd𝑦)) = (⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑦)))
4140oveqd 6810 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd𝑦))) = (𝑔(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑦))))
42 hofpropd.c . . . . . . . . . . 11 (𝜑𝐶 ∈ Cat)
4342ad3antrrr 709 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝐶 ∈ Cat)
4415ad2antrr 705 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd𝑥) ∈ (Base‘𝐶))
4526adantr 466 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
4645, 21syl6eqr 2823 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
4746eleq2d 2836 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ( ∈ ((Hom ‘𝐶)‘𝑥) ↔ ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))))
4847biimpa 462 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
49 simplrr 763 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
505, 6, 30, 43, 36, 44, 37, 48, 49catcocl 16553 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑦))) ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑦)))
5141, 50eqeltrd 2850 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd𝑦))) ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑦)))
525, 6, 30, 31, 32, 34, 35, 36, 37, 38, 51comfeqval 16575 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓) = ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓))
535, 6, 30, 31, 32, 34, 36, 44, 37, 48, 49comfeqval 16575 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑦))) = (𝑔(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐷)(2nd𝑦))))
5439oveq1d 6808 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑥(comp‘𝐷)(2nd𝑦)) = (⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐷)(2nd𝑦)))
5554oveqd 6810 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐷)(2nd𝑦))) = (𝑔(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐷)(2nd𝑦))))
5653, 41, 553eqtr4d 2815 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd𝑦))) = (𝑔(𝑥(comp‘𝐷)(2nd𝑦))))
5756oveq1d 6808 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓) = ((𝑔(𝑥(comp‘𝐷)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓))
5852, 57eqtrd 2805 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓) = ((𝑔(𝑥(comp‘𝐷)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓))
5929, 58mpteq12dva 4866 . . . . 5 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)) = ( ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓)))
6013, 19, 59mpt2eq123dva 6863 . . . 4 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))) = (𝑓 ∈ ((1st𝑦)(Hom ‘𝐷)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓))))
613, 4, 60mpt2eq123dva 6863 . . 3 (𝜑 → (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))) = (𝑥 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐷)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓)))))
621, 61opeq12d 4547 . 2 (𝜑 → ⟨(Homf𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))⟩ = ⟨(Homf𝐷), (𝑥 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐷)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓))))⟩)
63 eqid 2771 . . 3 (HomF𝐶) = (HomF𝐶)
6463, 42, 5, 6, 30hofval 17100 . 2 (𝜑 → (HomF𝐶) = ⟨(Homf𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))⟩)
65 eqid 2771 . . 3 (HomF𝐷) = (HomF𝐷)
66 hofpropd.d . . 3 (𝜑𝐷 ∈ Cat)
67 eqid 2771 . . 3 (Base‘𝐷) = (Base‘𝐷)
6865, 66, 67, 7, 31hofval 17100 . 2 (𝜑 → (HomF𝐷) = ⟨(Homf𝐷), (𝑥 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐷)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐷)(2nd𝑦))𝑓))))⟩)
6962, 64, 683eqtr4d 2815 1 (𝜑 → (HomF𝐶) = (HomF𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4322  cmpt 4863   × cxp 5247  cfv 6031  (class class class)co 6793  cmpt2 6795  1st c1st 7313  2nd c2nd 7314  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Homf chomf 16534  compfccomf 16535  HomFchof 17096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-cat 16536  df-homf 16538  df-comf 16539  df-hof 17098
This theorem is referenced by:  yonpropd  17116  oppcyon  17117
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